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Thermodynamics · with answers

Practice set

Try each question first, then open it for the answer. Each set links back to its part.

Learning path · Formula sheet · Practice set

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Part 1 · Thermal Equilibrium and the Zeroth Law

1. A thermometer reads 36 °C in body A and 36 °C in body B. A and B are then put in contact. Which way does heat flow?

Neither way: they are at the same temperature, so they are already in thermal equilibrium (zeroth law).

2. A and B are in thermal equilibrium, but B and C are not. Can A and C be in thermal equilibrium?

No. If A ~ C as well, then $T_{B} = T_{A} = T_{C}$ , so B ~ C, which contradicts the data.

3. Bodies A and B are each in thermal equilibrium with a third body C. Which quantity must A and B have in common? ((a) Heat content (b) Temperature (c) Internal energy (d) Mass)

Option (b): Temperature.

Part 2 · First Law of Thermodynamics

4. A gas absorbs 500 J of heat and its internal energy rises by 300 J. Find the work done by the gas.

$W = Q - Δ U = 200$ J.

5. Find the heat needed to raise 3 mol of a monatomic gas (

C_{v} = \frac{3}{2} R

) by 20 K at constant volume.

$Q = 3 \times 12.47 \times 20 \approx 748$ J.

6. 2 mol of an ideal gas is heated at constant pressure from 300 K to 350 K. Find the work done by the gas.

$W = n R Δ T = 2 \times 8.314 \times 50 \approx 831$ J.

7. The temperature of 4 mol of a diatomic gas (

f = 5

) falls from 400 K to 350 K. Find ΔU.

$Δ U = 2.5 \times 4 \times 8.314 \times (- 50) \approx - 4157$ J.

8. A gas absorbs 200 J of heat and does 80 J of work on its surroundings. Its internal energy: ((a) increases by 280 J (b) increases by 120 J (c) decreases by 120 J (d) increases by 200 J)

Option (b): increases by 120 J.

Part 3 · Indicator Diagrams, State Variables and Cycles

9. A gas expands from 2 m³ to 5 m³ at a constant pressure of 100 Pa. Find the work done by the gas.

$W = P Δ V = 300$ J.

10. From state A to B along path 1, a gas absorbs 80 J and does 30 J of work. Along path 2 it does 10 J of work. Find the heat absorbed along path 2.

$Δ U = 50$ J, so $Q_{2} = 50 + 10 = 60$ J.

11. A gas goes round a clockwise rectangular cycle between V = 1 and 3 m³ and P = 100 and 300 Pa. Find the net heat absorbed per cycle.

Area $= 2 \times 200 = 400$ J $= Q_{n e t}$ .

12. Over one cycle a gas absorbs 900 J of heat and rejects 600 J. Find the net work done by the gas.

$Δ U = 0$ , so $W_{n e t} = 900 - 600 = 300$ J.

13. A gas goes round a cycle that is anticlockwise on its P–V diagram and encloses an area of 50 J. Over one complete cycle: ((a) the gas does 50 J of work (b) the gas absorbs 50 J of net heat (c) the gas gives out 50 J of net heat (d) ΔU = −50 J)

Option (c): the gas gives out 50 J of net heat.

Part 4 · Molar Specific Heats: Cv, Cp and γ

14. Find

C_{p}

and γ for a gas with

C_{v} = 3 R

$C_{p} = 4 R$ , $γ = 4/3$ .

15. For a gas,

C_{p} = 29.1

J/mol·K. Find

C_{v}

$C_{v} = C_{p} - R \approx 20.8$ J/mol·K (a diatomic gas).

16. A gas has γ = 1.4. Find its degrees of freedom.

$\frac{f + 2}{f} = 1.4 \Rightarrow f = 5$ .

17. 1 mol of He (γ = 5/3) is mixed with 1 mol of O₂ (γ = 7/5). Find γ of the mixture.

$\frac{1}{2/3} + \frac{1}{2/5} = 4 = \frac{2}{γ - 1} \Rightarrow γ = 1.5$ .

18. For an ideal gas, C_p/C_v = 1.4. Its molar heat capacity at constant volume C_v is: ((a) 1.5R (b) 2.5R (c) 3.5R (d) 1.4R)

Option (b): 2.5R.

Part 5 · Isochoric, Isobaric and Isothermal Processes

19. At constant volume the pressure of a gas at 300 K is 2 atm. Find the pressure at 450 K.

$P / T$ constant: 3 atm.

20. 3 mol of a diatomic gas is heated at constant pressure from 280 K to 320 K. Find W, ΔU and Q.

$W = 3 \times 8.314 \times 40 \approx 998$ J, $Δ U \approx 2494$ J, $Q \approx 3492$ J.

21. 1 mol of ideal gas at 400 K is compressed isothermally from 10 L to 5 L. Find the work done by the gas and the heat exchanged.

$W = 8.314 \times 400 \times ln 0.5 \approx - 2305$ J; $Q = W$ : about 2305 J of heat is released.

22. 2 mol of an ideal gas expand isothermally at 300 K from 10 L to 20 L. The heat absorbed by the gas is about: ((a) 0 (b) 3458 J (c) 1502 J (d) 4988 J)

Option (b): 3458 J.

Part 6 · Adiabatic Process

23. Air (γ = 1.4) at 27 °C is compressed adiabatically to one-eighth of its volume. Find the final temperature.

$T_{2} = 300 \times 8^{0.4} \approx 689$ K (about 416 °C).

24. 2 mol of a monatomic gas expands adiabatically, cooling from 500 K to 300 K. Find the work done by the gas.

$W = \frac{n R ( T _{1} - T _{2} )}{γ - 1} = \frac{2 \times 8.314 \times 200}{2/3} \approx 4988$ J.

25. A gas at

1 0^{5}

Pa has an adiabatic bulk modulus of

1.4 \times 1 0^{5}

Pa. Find γ and its isothermal bulk modulus.

γ = 1.4; $B_{i so} = P = 1 0^{5}$ Pa.

26. A monatomic ideal gas at 300 K expands adiabatically to 8 times its volume. Its final temperature is: ((a) 75 K (b) 37.5 K (c) 9.4 K (d) 300 K)

Option (a): 75 K.

Part 7 · Free Expansion of a Gas

27. An ideal gas at 2 atm in an insulated vessel expands freely into an equal evacuated vessel. Find the final pressure and the change in temperature.

1 atm; no change in temperature.

28. In the free expansion of an ideal gas, which of Q, W, ΔU, P, V and T change?

Only P (falls) and V (rises). Q = W = ΔU = 0 and T is unchanged.

29. An ideal gas at 27 °C, in a thermally insulated vessel, expands freely into a vacuum until its volume doubles. Its final temperature is: ((a) 27 °C (b) 13.5 °C (c) −123 °C (d) −46 °C)

Option (a): 27 °C.

Part 8 · Polytropic Process

30. Find the molar heat capacity of a diatomic gas (γ = 7/5) in the process

P V^{1/2}

= constant.

$C = 2.5 R + \frac{R}{1 - 0.5} = 4.5 R$ .

31. A monatomic gas follows

P V^{1.3}

= constant. Find C and say what is unusual about it.

$C = 1.5 R - \frac{R}{0.3} \approx - 1.83 R \approx - 15.2$ J/mol·K: negative, so the gas cools while absorbing heat.

32. One mole of ideal gas expands along

P V^{2}

= constant from V to 2V, starting at 400 K. Find the final temperature.

$T \propto P V \propto V^{- 1}$ : 200 K.

33. One mole of a diatomic ideal gas (γ = 1.4) undergoes the process PV³ = constant. Its molar heat capacity in this process is: ((a) 2R (b) 3R (c) 2.5R (d) 0)

Option (a): 2R.

Part 9 · Reversible Processes, Heat Engines and Refrigerators

34. An engine of efficiency 30% absorbs 1500 J per cycle. Find the work done and the heat rejected per cycle.

W = 450 J; $Q_{2}$ = 1050 J.

35. A refrigerator with COP 5 removes 600 J per cycle from the cold space. Find the work needed and the heat rejected.

W = 120 J; $Q_{1}$ = 720 J.

36. Which is approximately reversible: (a) a block sliding to rest by friction, (b) very slow isothermal compression of a gas, (c) two gases mixing, (d) heat flowing from a hot body to a cold one?

(b).

37. A heat engine of efficiency 25% does 300 J of work per cycle. The heat it rejects per cycle is: ((a) 900 J (b) 1200 J (c) 75 J (d) 225 J)

Option (a): 900 J.

Part 10 · Second Law and the Carnot Cycle

38. A Carnot engine works between 127 °C and 27 °C. Find its efficiency.

$1 - 300/400 = 25%$ .

39. A Carnot engine of efficiency 40% rejects 600 J per cycle. Find the heat absorbed and the work done.

$Q_{2} / Q_{1} = 0.6$ : $Q_{1} = 1000$ J, W = 400 J.

40. A Carnot engine with its sink at 27 °C has an efficiency of 40%. To raise its efficiency to 50% with the same sink, the source temperature must be raised by: ((a) 100 K (b) 500 K (c) 600 K (d) 9 K)

Option (a): 100 K.